## Exploring Area/Perimeter Through Coordinate Geometry

### Joseph Furner PhD

#### Description

Students learn about area and perimeter through coordinate geometry. The use of children's literature, hands-on manipulatives, and the Internet will be incorporated.

#### Objectives

The student knows how to identify, locate, and plot ordered pairs of whole numbers on a graph or on the first quadrant of a coordinate system.

#### Materials

-Computer Lab with computers bookmarked with the Websites for the Internet Field Trip Lesson
-Book [Fly on the Ceiling: A Math Myth] by Julie Glass and Richard Walz---This book tells the history of the life of the famous French mathematician Rene Descartes (1596-1650) and how he developed the Cartesian Coordinate System. The story tells of Descartes experience with a fly and how he imagined it to land on the ceiling and saw the ceilingsas a coordinate plane. As the fly landed in a spot it represented an ordered pair coordinate. The children's book teaches the math concept of coordinate geometry.
-Large/clear geoboards and geobands with grid labeled 1-10 both x- and y-axes
-Internet Knowledge Sheets (see associated file)
-Measuring sticks or tape may be needed to measure the classroom for coordinate graphs
- 8.5 X 11 inch laminated dot matrix paper
- Wax crayons or water based transparency pens
- Unifix cubes and/or colored circle stickers to label coordinates

#### Preparations

1. Prior to the lesson the teacher can have the computers setup with the following Websites bookmarked:
Area and Perimeter Exploration at: www.shodor.org/interactivate/activities/coords2/index.html
Area and Perimeter Exploration at:http://www.math.com/school/subject3/practice/S3U1L2/S3U1L2Pract.html
Area and Perimeter Exploration at:http://www.mathgoodies.com/lessons/vol1/area_rectangle.html
Introduction to Coordinate Geomtery at: http://www.shodor.org/interactivate/discussions/fd2.html
More Intro. to Coordinate Geo. at: http://www.math.com/school/subject3/lessons/S3U1L2GL.html
Coordinate Geometry Grid Game at: http://www.shodor.org/interactivate/activities/coords2/index.html
Coordinate Geo. matching Workout practice at: http://www.math.com/school/subject3/practice/S3U1L2/S3U1L2Pract.html
Exploring Area and Perimeter at: http://www.geocities.com/jk02.geo/math6.html
Coorinate Grid to Print out to do Graphs at: http://www.kings.k12.ca.us/math/lessons/graphing/coordinate.html
Graphing Quiz at: http://www.kings.k12.ca.us/math/lessons/graphing/graphing.quiz.html
(Note that not all of these sites are used in the lesson, but may be used as follow-up activities to follow during the unit/week).
Too, previewing each weblink and setting the large number of bookmarks on student computers may take considerable time. It is suggested that a small cadre of student leaders be trained to accomplish the bookmarking task if possible.

2. Have enough geoboards available one for each student or use laminated dot matrix paper and pens/crayons

3. Copies of the Internet Field Trip Knowledge Worksheets should be duplicated.

4. Be sure to have all materials like the children's books, measuring tapes, worksheets, etc.

#### Procedures

NOTE: Students need the measurements of their bedrooms prior to beginning this lesson. For those who don't have them, create a "pretend" set for students to use.
**Prior to the lesson have the computers set up with the Websites bookmarked for easy student access.

1. Begin the lesson by reading the children's literature book, [The Fly on the Ceiling: A Math Myth] by Julie Glass and Richard Walz. The story tells of the life of the famous French mathematician Rene Descartes (1596-1650) and how he developed the Cartesian Coordinate System, or Coordinate Geometry, a connection of the branches of mathematics, algebra and geometry. After reading the book, ask students if they could imagine their bedroom ceilings or floors with a grid to locate flies or items in their rooms using ordered pairs. At this time, do some of the problems posed in the story.

2. Teach the students how to find points on a graph by explaining that they must use steps to find the points or address on the graph. For example, you must move 2 steps over from 0 and 3 steps up from that to find the point represented by the ordered pair (2,3) on a graph.

3. Introduce students to some terminology as it relates to the Cartesian (or Rectangular) Coordinate System also known as the Cartesian Plane. Students will need to learn about ordered pairs and how to graph them knowing about the x- and y-axes. In a set of ordered pairs (x,y), the x-coordinate is also called the abscissa and the y-coordinate is called the ordinate. Students will also know that the (0,0) ordered pair is called the origin or starting point for plotting or graphing points and lines. See more detailed instructions on using this method of introducing coordinate geometry at www.shodor.org/interactivate/discussions/fd2.html.

4. Pass out the geoboards and bands. Give the students several points to plot on the geoboards and have them put their fingers on the given point. {ie. (2,3), 5,1), (10,7), (3,9), (0,5)}. It may be best to use the larger clear goeboards which allow for more possible ordered pair combinations over the smaller 5 x 5 geoboards. Walk around the room to visually check that the students understand how to plot points. Explain to the class that they will be using coordinate geometry to learn about area and perimeter of rectangles. Too, dot matrixppaper can be produced using an AppleWorks 6 draw program. Laminating it can provided for reuse. Wax crayons or transparency pens can be used to plot and connect the points like a geoboard---this may be an alternative to the use of geoboards.

5. Give the students a set of 4 points to plot on their geoboards (make sure that when connected, these points will form a rectangle) ie. A=(2,3), B=(2,6), C=(6,6), andD= (6,3) and connect A to B to C to D to A, forming a rectangle. After the students have found the points, instruct them to connect all of the points using a band. The teacher may want to use unifix cubes or small circle stickers and label each ordered pair on the geoboard. Then have them find the perimeter of the rectangle by counting the spaces between the points. Ask them if they can think of an easier way to find the perimeter (count the spaces between the points for length and width, double it). They can derive the formula, perimeter equals two times length plus two times width.

6. Have the students find the area by counting the square units inside the rectangle. Ask them if they can think of an easier way to find the area (prompt as needed to ensure that students see the relationship between the length & width they counted for perimeter and the total number of spaces between points around the rectangle; if you multiply the length and the width together you get the area.) Practice with one or two more sets of points.

7. Together, guiding the students, have them go on an Internet Field Trip to use their new knowledge of coordinate geometry and perimeter & area of rectangles. Along with the students, guide them to read and review the Introduction to Coordinate Geomtery at: http://www.shodor.org/interactivate/discussions/fd2.html and More Intro. to Coordinate Geo. at: http://www.math.com/school/subject3/lessons/S3U1L2GL.html and discuss how this relates to the story about the fly on the ceiling.

8. Pass out the Internet Field Trip Assessment Knowledge Sheets (see associated file) for the students to fill out as they visit each Website. Go over directions and answer questions. Circulate as students work to monitor and provide assistance. (See Weblinks)

First stop: www.shodor.org/interactivate/activities/coords2/index.html The Geometry Grid Game: at this site has students find the x and y coordinates of the house on the grid. Students can also pick their own coordinates and have the house move there.

Second stop: http://www.math.com/school/subject3/practice/S3U1L2/S3U1L2Pract.html Coordinate Geometry Matching Workout: at this site the students match coordinates with descriptions of points on the graph. The computer will automatically check their answers.

Third Stop: http://www.mathgoodies.com/lessons/vol1/area_rectangle.html Exploring Area: this site gives a brief overview of area and then asks the students to calculate the area of several rectangles. The computer will check their answers.

9. Give students the following directions. You may need to write them on the board depending on your class. Directions include:
A. Make a coordinate graph that reflects the dimensions of either your bedroom or classroom. Be sure to:
-Title the graph
-Choose an appropriate scale

-Label each axes of your graph

-Plot and label ordered pair values

- To your best ability with known measurements, accurately place/label ordered pairs and calculate the area of the room and label items in your bedroom or classroom with ordered pair values representing the locations of such items. (Students may be given measuring tapes for accurate coordinate graphs.)

B. Use the graph to write two statements that state the ordered pair coordinates of your bedroom or classroom and then calculate the area and perimeter of the location with correct measurement units labeled. Be descriptive in how you arrived at each conclusion. (These will be assessed as FCAT Type Responses.)

10. Have students share their graphs with classmates and the teacher. Consider making a bulletin board of the graphs from this lesson activity to reinforce and make available to and provide information for students.

#### Assessments

The lesson offers opportunities for several forms of formative assessment.
1. Questions and answers after reading the children's literature book should be formatively assessed.
2. Performance-based assessment requiring students to make a coordinate graph reflecting the dimensions of their bedroom or classroom should be formatively assessed. Graphs should be correctly titled, scaled, and labeled.
3. Students should be formatively assessed through teacher observation while students work with geoboards.
4. Instant feedback is provided by games and tests at Weblink sites. The assessment provided at the Weblink sites is incredible. The teacher may find himself/herself fully engaged in a variety of fun activities and may find it difficult to break away! Many of the activities not only scored correct and incorrect answers, but also indicated elapsed time to complete tasks.
5. FCAT type assessment from Internet Field Trip Assessment Knowledge Worksheets with rubric/checklist should be assessed for each student.

Note: Circulate and formatively assess students as they use the technology tools. Provide assistance for students who are experiencing difficulty and monitor accordingly.

#### Extensions

Another children's literature book for this topic is: [Spaghetti and Meatballs for All: A Mathematical Story] by Marilyn Burns and Debbie Tilley. This book tells the story of a group of people deciding how to arrange enough square tables to all be able to dine out at their favorite restaurant. It teaches the concept of perimeter in the context of a story/problem.

Follow Up Games and Coordinate Grid to Printout at:
Coordinate Grid to Print out to do Graphs at: http://www.kings.k12.ca.us/math/lessons/graphing/coordinate.html
Graphing Quiz at: http://www.kings.k12.ca.us/math/lessons/graphing/graphing.quiz.html

Use masking tape to make a coordinate grid on the classroom floor to locate ordered pairs and calculate areas and perimeters of various items and areas.

Use the color tiles to explore more ideas with area and perimeter. For example: Use color tiles to explore the concepts of Constant Area and Constant Perimeter by creating array/areas with the color tiles for example 4 x 5 rectangle and explore its area and perimeter. Students can see more connections to the geoboards and the math concept at hand.

Use the Math Blaster Geometry Software to do more work as it relates to Coordinate Geometry, Area, and perimeter----they also have as part of this visual software geoboards.

This lesson could easily be turned into a unit or three day lesson as follows:
Perhaps the first day would be the introduction, the vocabulary, the practice on the geoboards, and some real-world problem. The second day would be the group instruction from the intro sites, the Internet Field Trip, and a discussion of a real-world problem using what they learned on the Internet. For homework, students would get the measurements needed from their bedrooms. On the third day, students would actually create the graphs of their rooms and ordered pairs for the furniture, and maybe do another real-world problem involving area or perimenter. (for example: How many square feet of carpet do you need for your bedroom? How many feet of wall paper border would you need? etc. ) Also, use real-world problems like: Juan's bedroom is 10X12. He has a bed that is 6X8, a dresser that is 3X2 and a bookcase that is 1.5X2.5. Draw a diagram of how he can arrange the furniture and still have room to walk around. Include the ordered pair coordinates for each piece of furniture. (This is just an example.)

Teachers should be encouraged to pick and choose from the collection to tailor the the delivery to their own students. This would help bring the lesson closer to the suggested time frame. The Website may also be used with other lessons within a unit as it relates to this topic.
Introduction to coordinate Geometry

Web supplement for Exploring Area/Perimeter Through Coord. Geometry
More Introduction to Coordinate Geometry

Web supplement for Exploring Area/Perimeter Through Coord. Geometry
Coordinate Geometry Grid Game

Web supplement for Exploring Area/Perimeter Through Coord. Geometry
Coordinate Geometry matching Workout practice

Web supplement for Exploring Area/Perimeter Through Coord. Geometry
Area and Perimeter Exploration 1

Web supplement for Exploring Area/Perimeter Through Coord. Geometry
Area and Perimeter Exploration 2

Web supplement for Exploring Area/Perimeter Through Coord. Geometry
Area and Perimeter Exploration 3